Past exam of the options II course of the University of Oxford / 2007 / B / B2 / ii

A primitive seventh root generates all seventh roots, so the splitting field is $\mathbb{Q}({\zeta }_7)$. Its minimal polynomial is the given irreducible cyclotomic polynomial $1+x+\cdots +x^6$. Therefore $\boxed{[K:\mathbb{Q}]=6}$.